On maximal spacelike submanifolds

In this paper, we first prove that the nonexistence of the maximal spacelike submanifolds with positive sectional curvture in the pseudo-Riemannian space form of negative sectional curvature, then give a condition under which the submanifolds are totally geodesic.I would like to thank prof. K. Ogiue for his advice and to thank also the referee for valuable suggestions.